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Gauge Theoretic Aspects of the Geometric Langlands Correspondence

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NORTHWESTERN UNIVERSITY Gauge Theoretic Aspects of the Geometric Langlands Correspondence A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS for the degree DOCTOR OF PHILOSOPHY Field of Mathematics By Chris Elliott EVANSTON, ILLINOIS June 2016 2 ABSTRACT Gauge Theoretic Aspects of the Geometric Langlands Correspondence Chris Elliott In their revolutionary 2006 paper, Kapustin and Witten described a fascinating bridge between geometric representation theory and the quantum theory of supersymmetric gauge fields. They explained how, by performing a suitable topological twist, one can obtain categories of sheaves on moduli stacks of holomorphic and flat G-bundles as categories of boundary conditions in supersym- metric gauge theories, and why the physical phenomenon of S-duality should yield a conjectural equivalence of categories known as the geometric Langlands correspondence. In this thesis, I begin to make some of the structures introduced by Kapustin-Witten and other theoretical physicists mathematically rigorous, with the eventual aim of systematically using the huge amount of struc- ture possessed by the panoply of supersymmetric gauge theories in the theoretical physics literature to draw new insights about geometric representation theory. The present work consists of two dis- tinct approaches. Firstly I give a construction of a generalization of abelian gauge theories using the mathematical structure of a factorization algebra, and explain how S-duality for these theories can be described as a version of the Fourier transform. Then, I explain how to construct classical supersymmetric gauge theories using derived algebraic geometry, introduce an appropriate notion of twisting for such theories, and prove that the twists introduced by Kapustin and Witten yield the moduli stacks of interest for the geometric Langlands correspondence. 3 Acknowledgements I would like to begin by crediting Philsang Yoo, with whom a significant part of this research was jointly carried out in a productive and on-going collaboration. The material in chapters 7 to 10 and the appendices of this thesis is from our joint preprint \Geometric Langlands Twists of N = 4 Gauge Theory from Derived Algebraic Geometry" [EY15], and the ideas I discuss in chapter 13 are part of joint work currently in progress. It's been a great pleasure to work with Philsang; here's to many more years of productive collaboration! I would like to thank all of those people who've supported me through graduate school, without whom none of this could have been accomplished. First and foremost I'd like to thank my advisors: first David Nadler, then Kevin Costello. Although we've been geographically separated for some time now, their knowledge, advice and encouragement have been invaluable throughout my time at Northwestern. I'd also like to single out some good friends who listened to me and helped me through every stage of my work. Saul Glasman is one of my oldest friends, and his steadfast support and encouragement, and boundless enthusiasm, were and still are an incredible source of inspiration. Thel Seraphim helped me come up with some of the ideas that started this work off, and gave me invaluable advice and constructive criticism about all manner of things. I hope we meet again soon! During this research I've had great conversations and received explanations and advice from many other mathematicians. I'd particularly like to thank David Ben-Zvi, Damien Calaque, Ryan Grady, Michael Gr¨ochenig, Sam Gunningham, Owen Gwilliam, Boris Hanin, Aron Heleodoro, 4 Theo Johnson-Freyd, Clemens Koppensteiner, Kobi Kremnitzer, Ian Le, Toly Preygel, Nick Rozen- blyum, Pavel Safronov, Claudia Scheimbauer, Jack Shotton, Brian Williams, and Jesse Wolfson for discussions that directly helped me with the work presented here. Outside of mathematics, my friends in Chicago and elsewhere have always been there for me with drinks, kind words and affection. To name just a few, I'd like to thank Cheryl and Brendan (and Bailey), Carl, Leah and Nathan, Joel, Paul, Erik, Kym, Ness, the holiday crowd (especially Nick and Rosie), Chris and Liz, Laurel and Hannah for all the good times we've shared together. I'd also like to thank Hugo for always being there when I've needed him, and always having something apt to say. Of course, I could never have made it to where I am today without a lifetime of support from my family. Thank you for everything you've done for me, and for standing by me in all my choices. Finally, and above all, I'd like to thank Claire, without whose love I don't know where I would be. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development & Innovation. 5 Table of Contents ABSTRACT 2 Acknowledgements 3 List of Figures 8 List of Tables 9 Chapter 1. Introduction 10 1.1. Introduction to S-Duality 10 1.2. Introduction to the Geometric Langlands Program 24 1.3. The Approach of Kapustin and Witten 38 1.4. Outline of this Thesis 48 1.5. Conventions 52 Part 1. Abelian Duality 56 Chapter 2. Introduction and Motivation 57 Chapter 3. The BV Formalism for Free Field Theories 59 3.1. The Idea of the BV Formalism 59 3.2. Derived Spaces from Cochain Complexes 61 3.3. The Action Functional and the Classical Factorization Space 65 3.4. Quantization of Free Factorization Algebras 69 3.5. Smearing Observables 71 Chapter 4. Generalized Maxwell Theories as Factorization Algebras 76 6 4.1. Generalized Maxwell Theories 76 4.2. Free Theories from p-forms 79 Chapter 5. Expectation Values 83 5.1. Expectation Values from Free Quantum Factorization Algebras 83 5.2. Computing Expectation Values 86 Chapter 6. Fourier Duality for Polynomial Observables 91 6.1. Feynman Diagrams for Fourier Duality 91 6.2. Fourier Duality and Expectation Values 97 6.3. Wilson and `t Hooft Operators 101 Part 2. Twists of N = 4 Supersymmetric Gauge Theories 106 Chapter 7. Introduction 107 Chapter 8. Classical N = 4 Theories and their Twists 109 8.1. Holomorphic and Topological Twists 109 8.2. Twisted Supersymmetric Field Theories 120 Chapter 9. Constructing Supersymmetric Gauge Theories 144 9.1. Compactification and Dimensional Reduction 144 9.2. N = 1 Super-Yang-Mills in Ten Dimensions 146 9.3. Twistor Space Formalism 149 9.4. Holomorphic Chern-Simons Theory on Super Twistor Space 152 Chapter 10. Equations of Motion in the Twisted Theories 162 10.1. The Holomorphic Twist 162 10.2. The B-twist 169 10.3. The A-twist as a Limit of Holomorphic-Topological Twists 178 Part 3. Outlook and Future Work 191 7 Chapter 11. Abelian Duality for N = 4 Theories 192 Chapter 12. Constructing Topological Field Theories 194 Chapter 13. Vacua and Singular Support Conditions 198 Chapter 14. N = 2 Theories and the Future 202 References 204 Appendices 215 Appendix A. Supersymmetry Algebras 215 Appendix B. Lie Algebras and Deformation Theory 224 8 List of Figures 5.1 Example of a Feynman diagram 88 6.1 Example Feynman diagram for a dual observable 92 6.2 Feynman diagram for a double dual observable 94 6.3 Illustration in the proof that observables are equal to their double duals 94 9 List of Tables 1.1 Masses and charges of electrically and magnetically charged states 17 1.2 Table of groups and their Langlands duals 18 10 CHAPTER 1 Introduction 1.1. Introduction to S-Duality We will begin this thesis somewhat philosophically, with a discussion of what it should mean for a pair of quantum field theories to be \dual". Roughly speaking, when physicists speak about a duality, they mean a pair of prequantum field theories (by which I mean classical field theories equipped with some additional data, including a specific action functional with specified values for coupling constants, that determines a choice of quantization) along with an equivalence of their quantizations. For example, consider the following definition of duality, from a physical point of view. 0 0 0 Definition 1.1.1. [Tes16] A pair of theories (Φ; τ; Sτ ) and (Φ ; τ ;Sτ 0 ) given by a space of fields and an action functional depending on some auxilliary parameters τ or τ 0 are dual if there exists 0 a moduli space M of quantum field theories having boundary points z0 and z0, coordinates τ0 and 0 0 0 0 τ0 on M near these boundary points, and maps f : Obs !O(Φ) and f : Obs !O(Φ ) from the 0 local observables near z0 and z0 such that there exist equivalences of asymptotic expansions Z ∼ −Sτ (φ) hOiz = Dφe 0 f(O)(φ, τ0) Z −S0 (φ0) ∼ 0 τ0 0 0 0 and hOiz = Dφ e 0 f (O)(φ ; τ0) 0 near z0 and z0 respectively. Making this definition precise relies on a number of things, not least an appropriate definition of a quantum field theory and an appropriate definition for the path integral expressions given on the 11 right hand of the above equivalences. Nevertheless it captures the idea for what sort of thing a duality should be: a pair of apparently distinct prequantum field theories which nonetheless, when quantized, describe equivalent physics. Taking the non-rigorous parts for granted, definition 1.1.1 admits a neat simplification in the case where the moduli space M is a single point. 0 0 0 Example 1.1.2. A pair of prequantum theories (Φ; τ; Sτ ) and (Φ ; τ ;Sτ 0 ) are dual if there is a correspondence Obs f f 0 { # ObsΦ ObsΦ0 0 where ObsΦ and ObsΦ0 are quantizations of the two classical theories, and where f(O) and f (O) have the same expectation value for each observable O 2 Obs.
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